Generalized Gr\"otzsch Graphs

TL;DR

This paper introduces two new families of graphs, $G_m$ and $H_m$, generalizing Gr"otzsch graphs, with properties like non-planarity, triangle-freeness, Hamiltonicity, and specific chromatic numbers, expanding understanding of triangle-free graphs.

Contribution

The paper constructs and analyzes two new families of graphs, $G_m$ and $H_m$, generalizing Gr"otzsch graphs with detailed properties and comparisons to existing constructions.

Findings

$G_m$ is 4-chromatic and non-planar.

$H_m$ is 3-chromatic and non-planar.

Both families are triangle-free and Hamiltonian.

Abstract

The aim of this paper is to present a generalization of Gr\"otzsch graph. Inspired by structure of the Gr\"otzsch's graph, we present constructions of two families of graphs, $G_m$ and $H_m$ for odd and even values of $m$ respectively and on $n = 2m +1$ vertices. We show that each member of this family is non-planar, triangle-free, and Hamiltonian. Further, when $m$ is odd the graph $G_m$ is maximal triangle-free, and when $m$ is even, the addition of exactly $\frac{m}{2}$ edges makes the graph $H_m$ maximal triangle-free. We show that $G_m$ is 4-chromatic and $H_m$ is 3-chromatic for all $m$. Further, we note some other properties of these graphs and compare with Mycielski's construction.